Finite Group Databasebeta

← C₂×C₇⋊(C₂₃²⋊C₃) #36372425

Subgroups — C₂×C₇⋊(C₂₃²⋊C₃) #36372425

Order 22218.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇⋊(C₂₃²⋊C₃)	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₇×C₂₃²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₂₃²	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₂₃²⋊C₃	—	7	1	no	maximal, Hall
5	C₂₃²⋊C₃	—	7	1	no	Hall
6	C₂×C₂₃²	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
7	C₂₃²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
8	C₂×C₇×C₂₃	—	3	8	no	—
9	C₇×C₂₃	—	3	8	no	—
10	C₂×C₂₃	—	3	8	no	—
11	C₂×ΓL(1,𝔽₈)	—	529	1	no	maximal, Hall
12	C₂₃	—	3	8	no	—
13	ΓL(1,𝔽₈)	—	529	1	no	Hall
14	C₂×C₇	C₂₃²⋊C₃	1	1	yes	characteristic, complemented, Hall
15	C₇	C₂×C₂₃²⋊C₃	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₃	—	3703	1	no	Hall
17	C₃	—	3703	1	no	Sylow
18	C₂	C₇⋊(C₂₃²⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂	C₇⋊(C₂₃²⋊C₃)
commutator, cosocle, hypocenter	C₇×C₂₃²	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₂₃²	C₃
LCS	G ▷ C₇×C₂₃²
UCS	C₁ ◁ C₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇⋊(C₂₃²⋊C₃)	C₂	direct (N × H)
2. C₂×C₇×C₂₃²	C₃	semidirect
3. C₇×C₂₃²	C₂×C₃	semidirect
6. C₂×C₂₃²	ΓL(1,𝔽₈)	semidirect
7. C₂₃²	C₂×ΓL(1,𝔽₈)	semidirect
14. C₂×C₇	C₂₃²⋊C₃	semidirect
15. C₇	C₂×C₂₃²⋊C₃	semidirect
18. C₂	C₇⋊(C₂₃²⋊C₃)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇⋊(C₂₃²⋊C₃)	2
C₂×C₇×C₂₃²	3
C₂×C₂₃²⋊C₃	7
C₂×ΓL(1,𝔽₈)	529